A144698 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(P_{n}(x) = \left(4 + 3\,x + n + n x\right)\,P_{n-1}(x) + \left(-x - x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	5	4														9
2	30	45	16													91
3	210	450	305	64												1029
4	1680	4620	4530	1845	256											12931
5	15120	50400	64050	38250	10505	1024										179349
6	151200	589680	915600	716100	292530	57645	4096									2726851
7	1663200	7408800	13547520	13053600	7060410	2101050	308705	16384								45159669
8	19958400	99792000	209563200	239410080	160912080	64130220	14458530	1625445	65536							809915491
9	259459200	1437004800	3402907200	4489732800	3602899440	1800338400	550121250	96503850	8454905	262144						15647683989
10	3632428800	22054032000	58078944000	86819040000	80926322400	48629140560	18803677200	4526375700	629752530	43585245	1048576					324144347011
11	54486432000	359610451200	1041728688000	1738775808000	1845198986400	1295554629600	606895359840	186640423200	36092897610	4040026650	223169105	4194304				7169251065909
12	871782912000	6211453248000	19618747948800	36146558448000	43007697532800	34548289776000	19024499894400	7141822047360	1781866566480	280930469820	25579174530	1136817045	16777216			168660381612451
13	14820309504000	113331778560000	387453017952000	780714289555200	1029005327750400	930671847705600	589300686374400	261663219417600	80313197524560	16502376290400	2145567510450	160295949450	5767971305	67108864		4206087749174229
14	266765571072000	2178585497088000	8012556744192000	17522509612608000	25336074642278400	25475803280928000	18239670408960000	9360602963712000	3419764169421600	871340164334640	149183550406800	16141044219300	996383524530	29175400845	268435456	110850023476581571

Technical details

Node id: n-8fd7e451a6f8a0e6 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-0a560bfbcc45e3fa

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         5,          4;
  2:        30,         45,         16;
  3:       210,        450,        305,         64;
  4:      1680,       4620,       4530,       1845,        256;
  5:     15120,      50400,      64050,      38250,      10505,       1024;
  6:    151200,     589680,     915600,     716100,     292530,      57645,      4096;
  7:   1663200,    7408800,   13547520,   13053600,    7060410,    2101050,    308705,    16384;
  8:  19958400,   99792000,  209563200,  239410080,  160912080,   64130220,  14458530,  1625445,   65536;
  9: 259459200, 1437004800, 3402907200, 4489732800, 3602899440, 1800338400, 550121250, 96503850, 8454905, 262144;

%F Formula

The row polynomials satisfy P(n,x) = (n+4+(n+3)*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n + 4 + (n + 3)*x)*P[n - 1, x] - (x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (n + 4 + (n + 3)*x)*v[i - 1] - (x + x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle of 4-Eulerian numbers.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003